ASSET PRICING
Syllabus
EN
IT
Learning Objectives
LEARNING OUTCOMES:
The objective of this course is to understand the main mathematical techniques for the
modelling and the analysis of financial markets.
KNOWLEDGE AND UNDERSTANDING:
The course discusses the main stochastic methods for modelling financial markets and
pricing simple financial derivatives.
APPLYING KNOWLEDGE AND UNDERSTANDING:
Students will be able to describe in mathematical terms basic models for financial markets.
In particular they will drow their attention to the concept of market price of risk market
completeness and absence of arbitrage.
Students will also be able to apply quantitative techniques for evaluating different types of
financial derivatives.
MAKING JUDGEMENTS:
The course contains proofs of theorems and analytic properties. These aspects allows
students to build and develop logic arguments with a clear identification of hypotheses and
theses, and identify wrong or incorrect reasoning. The discussion of examples and model in
actuarial science will permit the students to understand the main characteristics of a
reasonable mathematical modelling of some phenomena.
COMMUNICATION SKILLS:
Students must be able to describe with the due scientific rigour, mathematical models for
representing longevity risk and discuss the necessary techniques for the valutaion of life
insurance contracts. This course provides all the instruments for communicating rigorously
quantitative results in the actuatial framework.
LEARNING SKILLS:
The course provides basic instruments for the development of further studies in the
actuarial framework. The more theoretical part allows students to independently face new
and more complex problems.
The objective of this course is to understand the main mathematical techniques for the
modelling and the analysis of financial markets.
KNOWLEDGE AND UNDERSTANDING:
The course discusses the main stochastic methods for modelling financial markets and
pricing simple financial derivatives.
APPLYING KNOWLEDGE AND UNDERSTANDING:
Students will be able to describe in mathematical terms basic models for financial markets.
In particular they will drow their attention to the concept of market price of risk market
completeness and absence of arbitrage.
Students will also be able to apply quantitative techniques for evaluating different types of
financial derivatives.
MAKING JUDGEMENTS:
The course contains proofs of theorems and analytic properties. These aspects allows
students to build and develop logic arguments with a clear identification of hypotheses and
theses, and identify wrong or incorrect reasoning. The discussion of examples and model in
actuarial science will permit the students to understand the main characteristics of a
reasonable mathematical modelling of some phenomena.
COMMUNICATION SKILLS:
Students must be able to describe with the due scientific rigour, mathematical models for
representing longevity risk and discuss the necessary techniques for the valutaion of life
insurance contracts. This course provides all the instruments for communicating rigorously
quantitative results in the actuatial framework.
LEARNING SKILLS:
The course provides basic instruments for the development of further studies in the
actuarial framework. The more theoretical part allows students to independently face new
and more complex problems.
Prerequisites
Financial mathematics (compound intereset rates, force of interest, interest rate curve, bonds).
Probability (Discrete and continuous random variables, expectations, conditional probability)
Main definitions and features of financial derivatives.
Probability (Discrete and continuous random variables, expectations, conditional probability)
Main definitions and features of financial derivatives.
Program
Week 1 - Week 2: Financial Markets and Arbitrage and One-Period Model
a) Financial derivatives: forwards, options, and swaps
b) Arbitrage and replication
c) Fundamental pricing idea
d) The One-Period Model
e) Portfolio value and arbitrage
f) Risk-neutral pricing
g) Market completeness
Objective: Understand the no-arbitrage principle and basic financial instruments. Analyze simple models of arbitrage and complete markets in discrete time.
Week 3 - The Multi-Period Model
a) Martingales and adapted processes
b) No-arbitrage and martingale measures
c) Pricing by replication
Objective: Extend discrete-time pricing to multiple periods and understand its connection to martingale theory.
Week 4 - Probability and Stochastic Processes
a) Filtrations conditional expectation
b) Martingales
c) Properties and construction of Brownian motion
d) Quadratic variation and Introduction to Itô integrals
e) Itô’s formula
Objective: Use stochastic calculus to analyze dynamics of financial processes.
Week 5 - The Black-Scholes Model
a) Black-Scholes PDE and formula
b) Hedging strategies
c) The Greeks
Objective: Apply the Black-Scholes model to price and hedge European derivatives.
Week 6 - General Arbitrage Pricing Theory
a) Fundamental Theorems of Asset Pricing
b) Market completeness and replication
c) More general models for financial markets
Objective: Understand the theoretical underpinnings of modern asset pricing.
a) Financial derivatives: forwards, options, and swaps
b) Arbitrage and replication
c) Fundamental pricing idea
d) The One-Period Model
e) Portfolio value and arbitrage
f) Risk-neutral pricing
g) Market completeness
Objective: Understand the no-arbitrage principle and basic financial instruments. Analyze simple models of arbitrage and complete markets in discrete time.
Week 3 - The Multi-Period Model
a) Martingales and adapted processes
b) No-arbitrage and martingale measures
c) Pricing by replication
Objective: Extend discrete-time pricing to multiple periods and understand its connection to martingale theory.
Week 4 - Probability and Stochastic Processes
a) Filtrations conditional expectation
b) Martingales
c) Properties and construction of Brownian motion
d) Quadratic variation and Introduction to Itô integrals
e) Itô’s formula
Objective: Use stochastic calculus to analyze dynamics of financial processes.
Week 5 - The Black-Scholes Model
a) Black-Scholes PDE and formula
b) Hedging strategies
c) The Greeks
Objective: Apply the Black-Scholes model to price and hedge European derivatives.
Week 6 - General Arbitrage Pricing Theory
a) Fundamental Theorems of Asset Pricing
b) Market completeness and replication
c) More general models for financial markets
Objective: Understand the theoretical underpinnings of modern asset pricing.
Books
1. T. Bjork. Arbitrage theory in continuous time. Oxford University Press.
2. J. Cochrane. Asset pricing. Princeton University Press
2. J. Cochrane. Asset pricing. Princeton University Press
Bibliography
1. T. Bjork. Arbitrage theory in continuous time. Oxford University Press.
2. J. Cochrane. Asset pricing. Princeton University Press
2. J. Cochrane. Asset pricing. Princeton University Press
Teaching methods
Theoretical lectures and exercises.
Exam Rules
Learning will be verified through a written exam where you will be asked to solve exercises and to answer theoretical questions on all topics covered during letures.
The exam is passed if the written test is evaluated 18/30 or more. If the exam is passed (i.e. the mark in the written test is at least 18/30), you can withdraw and repeat the exam in one of the next exam calls. This can be done ONE TIME ONLY. The mark obtained at the next exam date cancels the previous mark and cannot be rejected.
To pass the exam, the student must demonstrate the ability to discuss the condition of no-arbitrage and market completeness and use these characteristics to determine the pricing of derivatives and hedging strategies. Moreover the student must be able to describe the feature of mathematical models for the description of financial markets and use to use these feature to solve the problems of pricing and risk hedging. The student must be able to provide both economic-financial and mathematical justifications for their statements.
Criteria for Formulating the Grade Out of Thirty:
- Not Sufficient: Significant deficiencies and/or inaccuracies in the knowledge and understanding of the topics; limited analytical and synthesis skills, frequent generalizations.
- 18-20: Barely sufficient knowledge and understanding of the topics with possible imperfections; sufficient analytical, synthesis, and judgment autonomy skills.
- 21-23: Routine knowledge and understanding of the topics; correct analytical and synthesis skills with coherent logical argumentation.
- 24-26: Fair knowledge and understanding of the topics; good analytical and synthesis skills with arguments expressed with sufficient mathematical and economic/financial rigor.
- 27-29: Complete knowledge and understanding of the topics; considerable analytical and synthesis skills. Good judgment autonomy and arguments expressed with good mathematical and economic/financial rigor.
- 30-30L: Excellent level of knowledge and understanding of the topics. Remarkable analytical, synthesis, and judgment autonomy skills. Arguments expressed with excellent mathematical and economic/financial mastery.
The exam is passed if the written test is evaluated 18/30 or more. If the exam is passed (i.e. the mark in the written test is at least 18/30), you can withdraw and repeat the exam in one of the next exam calls. This can be done ONE TIME ONLY. The mark obtained at the next exam date cancels the previous mark and cannot be rejected.
To pass the exam, the student must demonstrate the ability to discuss the condition of no-arbitrage and market completeness and use these characteristics to determine the pricing of derivatives and hedging strategies. Moreover the student must be able to describe the feature of mathematical models for the description of financial markets and use to use these feature to solve the problems of pricing and risk hedging. The student must be able to provide both economic-financial and mathematical justifications for their statements.
Criteria for Formulating the Grade Out of Thirty:
- Not Sufficient: Significant deficiencies and/or inaccuracies in the knowledge and understanding of the topics; limited analytical and synthesis skills, frequent generalizations.
- 18-20: Barely sufficient knowledge and understanding of the topics with possible imperfections; sufficient analytical, synthesis, and judgment autonomy skills.
- 21-23: Routine knowledge and understanding of the topics; correct analytical and synthesis skills with coherent logical argumentation.
- 24-26: Fair knowledge and understanding of the topics; good analytical and synthesis skills with arguments expressed with sufficient mathematical and economic/financial rigor.
- 27-29: Complete knowledge and understanding of the topics; considerable analytical and synthesis skills. Good judgment autonomy and arguments expressed with good mathematical and economic/financial rigor.
- 30-30L: Excellent level of knowledge and understanding of the topics. Remarkable analytical, synthesis, and judgment autonomy skills. Arguments expressed with excellent mathematical and economic/financial mastery.